Gravity, Quantum Fields & Information (GQFI) Group Calendar
Group activities, such as seminars and journal clubs, will be announced in the calendar below:
Past seminars:
24.11.17 - Roman Orus, "Overview: from (many) qubits to space-time"
In this talk I will make an overview of how space-time properties emerge from the entanglement structure of many-body wavefunctions. I will mainly focus on the connection between Entanglement Renormalization and AdS/CFT, but I will mention briefly other topics such as the appearance of spin networks in symmetric tensor networks, and the definition of "entanglement Hamiltonians" through a bulk-boundary correspondence for Projected Entangled Pair States. I will also discuss several open questions along these directions.
Many quantities in conformal field theory, most notably entanglement entropies, can only be defined with reference to a UV cutoff. The cutoff can be chosen and adjusted depending on one's purposes; it represents a sort of gauge freedom. I will explain that this gauge freedom leads to physical consequences in a way that closely mirrors the construction of the Berry phase. In Berry's language, the control space is the kinematic space (the space of pairs of points) in a CFT and the changing Hamiltonians are the modular Hamiltonians of CFT subregions. In CFT2, the modular Berry "phase" is a shift in the normalization of OPE blocks, which is (the exponential of) the differential entropy computed on the path taken in the (kinematic) control space. All recently reported applications of integral geometry to the AdS/CFT correspondence are most naturally understood in this language. For example, conditional entropy is the Berry connection while the Crofton form ("the density of geodesics in AdS3") is the Berry curvature. I will sketch a few generalizations and potential applications of the modular Berry connection that go beyond the ground state of a CFT2.
17.11.17 - Karl Jansen, "The Quest for Solving Quantum Chromodynamics: the tensor network approach"
The strong interaction of quarks and gluons is described theoretically within the framework of Quantum Chromodynamics (QCD). The most promising way to evaluate QCD for all energy ranges is to formulate the theory on a 4-dimensional Euclidean space-time grid, which allows for numerical simulations on state of the art supercomputers. We will review the status of lattice QCD calculations providing examples such as the hadron spectrum and the inner structure of nucleons. We will then point to problems that cannot be solved by conventional Monte Carlo simulation techniques, i.e. chemical potentials and understanding the violation of charge and parity symmetry. It is demonstrated at the example of the Schwinger model that tensor network techniques are able to overcome these problems opening thus a possible path for a solution also in QCD.
15.11.17 - Matt Headrick, "Riemannian and Lorentzian flow-cut theorems"
We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut theorem for boundary regions, applied recently to develop a "bit-thread" interpretation of holographic entanglement entropies. We also prove various properties of the max flow and min cut, including respective nesting properties. In the Lorentzian setting, we prove the analogous min flow-max cut theorem, which states that the volume of a maximal slice equals the flux of a minimal flow, where a flow is defined as a divergenceless timelike vector field with norm at least 1. This theorem includes as a special case a continuum version of Dilworth's theorem from the theory of partially ordered sets. We include a brief review of the necessary tools from the theory of convex optimization, in particular Lagrangian duality and convex relaxation.
20.10.17 - Sukhi Singh, "Tensor networks, symmetries, and holography"
Ground states of local hamiltonians on a quantum lattice can often be represented efficiently by means of a tensor network. In particular, the ground state of a quantum critical system -- described by a conformal field theory (CFT) in the continuum -- can be efficiently represented by a certain tensor network known as the MERA. According to a recent conjecture, the MERA could provide a discrete realization of at least certain aspects of the AdS/CFT correspondence. Several qualitative similarities between the MERA and the AdS/CFT correspondence have been observed, though how the MERA representation of a critical ground state could possibly encode a dual bulk description in one higher dimension is still an open question. In this talk, I will review the MERA-AdS/CFT conjecture and introduce a candidate bulk description of the MERA. This bulk description has some interesting features, some of which are reminiscent of AdS/CFT. In particular, I will show how this bulk description gauges an onsite global boundary symmetry in the bulk, a general feature of AdS/CFT.
19.10.17 - Wilke van der Schee, "Entanglement, MERA and modular hamiltonians"
18.10.17 - Netta Engelhardt, "Decoding the Apparent Horizon: A Coarse-Grained Holographic Entropy"
When a black hole forms from collapse in a holographic theory, the information in the black hole interior remains encoded in the boundary. We prove that the area of the black hole's apparent horizon is precisely the entropy associated to coarse graining over the information in its interior, subject to knowing the exterior geometry. This is the maximum holographic entanglement entropy that is compatible with all classical measurements conducted outside of the apparent horizon. We identify the boundary dual to this entropy and explain why it obeys a Second Law of Thermodynamics.
18.10.17 - William Cottrell, "Complexity is Simple"
In this note we investigate the role of Lloyd's computational bound in holographic complexity. Our goal is to translate the assumptions behind Lloyd's proof into the bulk language. In particular, we discuss the distinction between orthogonalizing and `simple' gates and argue that these notions are useful for diagnosing holographic complexity. We show that large black holes constructed from series circuits necessarily employ simple gates, and thus do not satisfy Lloyd's assumptions. We also estimate the degree of parallel processing required in this case for elementary gates to orthogonalize. Finally, we show that for small black holes at fixed chemical potential, the orthogonalization condition is satisfied near the phase transition, supporting a possible argument for the Weak Gravity Conjecture first advocated in Brown et al.
16.10.17 - Phuc Nguyen, "Entanglement of purification: from spin chains to holography"
Purification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of many degrees of freedom, one natural purification is the one with minimal entanglement. Here we study the entropy of the minimally entangled purification, called the entanglement of purification, in three model systems: an Ising spin chain, conformal field theories holographically dual to Einstein gravity, and random stabilizer tensor networks. We conjecture values for the entanglement of purification in all these models, and we support our conjectures with a variety of numerical and analytical results. We find that such minimally entangled purifications have a number of applications, from enhancing entanglement-based tensor network methods for describing mixed states to elucidating novel aspects of the emergence of geometry from entanglement in the AdS/CFT correspondence.
13.10.17 - Johannes Knaute, "Entanglement Entropy in holographic QCD phase diagrams"
We calculate the holographic entanglement entropy for the holographic QCD phase diagram considered in [arXiv:1702.06731] and explore the resulting qualitative behavior over the temperature-chemical potential plane. In agreement with the thermodynamic result, the phase diagram exhibits the same critical point as the onset of a first-order phase transition curve. We compare the phase diagram of the entanglement entropy to that of the thermodynamic entropy density and find a striking agreement in the vicinity of the critical point. Thus, the holographic entanglement entropy qualifies to characterize different phase structures. The scaling behavior near the critical point is analyzed through the calculation of critical exponents.
12.10.17 - Christian Northe, "Topological Complexity in AdS3/CFT2"
We consider subregion complexity within the AdS3/CFT2 correspondence. We rewrite the volume proposal, according to which the complexity of a reduced density matrix is given by the spacetime volume contained inside the associated Ryu-Takayanagi (RT) surface, in terms of an integral over the curvature. Using the Gauss-Bonnet theorem we evaluate this quantity for specific examples. In particular, we find a discontinuity when there is a change in the RT surface, given by a topological contribution. There is no further temperature dependence of the subregion complexity. We propose a CFT expression for this complexity based on kinematic space, and use it to reproduce some of our explicit gravity results obtained at zero temperature.
09.10.17 - Lucas Hackl, "Entanglement and Chaos: Linear entropy production at instabilities"
The rate of entropy production in a classical dynamical system is characterized by the Kolmogorov-Sinai entropy rate given by the sum over all positive Lyapunov exponents of the system. I will prove a quantum version of this result valid for bosonic systems with unstable quadratic Hamiltonian: the entanglement entropy of squeezed coherent states grows linearly for large times, with a rate determined by the Lyapunov exponents and the choice of the subsystem. I will discuss its application to quantum field theory and explain our conjecture on what this result implies for quantum chaos and thermalization in systems with periodic orbits.
Gravity, Quantum Fields & Information (GQFI) Group Calendar
Group activities, such as seminars and journal clubs, will be announced in the calendar below:
Past seminars:
24.11.17 - Roman Orus, "Overview: from (many) qubits to space-time"
In this talk I will make an overview of how space-time properties emerge from the entanglement structure of many-body wavefunctions. I will mainly focus on the connection between Entanglement Renormalization and AdS/CFT, but I will mention briefly other topics such as the appearance of spin networks in symmetric tensor networks, and the definition of "entanglement Hamiltonians" through a bulk-boundary correspondence for Projected Entangled Pair States. I will also discuss several open questions along these directions.17.11.17 - Bartek Czech, "Modular Berry Connection"
Many quantities in conformal field theory, most notably entanglement entropies, can only be defined with reference to a UV cutoff. The cutoff can be chosen and adjusted depending on one's purposes; it represents a sort of gauge freedom. I will explain that this gauge freedom leads to physical consequences in a way that closely mirrors the construction of the Berry phase. In Berry's language, the control space is the kinematic space (the space of pairs of points) in a CFT and the changing Hamiltonians are the modular Hamiltonians of CFT subregions. In CFT2, the modular Berry "phase" is a shift in the normalization of OPE blocks, which is (the exponential of) the differential entropy computed on the path taken in the (kinematic) control space. All recently reported applications of integral geometry to the AdS/CFT correspondence are most naturally understood in this language. For example, conditional entropy is the Berry connection while the Crofton form ("the density of geodesics in AdS3") is the Berry curvature. I will sketch a few generalizations and potential applications of the modular Berry connection that go beyond the ground state of a CFT2.17.11.17 - Karl Jansen, "The Quest for Solving Quantum Chromodynamics: the tensor network approach"
The strong interaction of quarks and gluons is described theoretically within the framework of Quantum Chromodynamics (QCD). The most promising way to evaluate QCD for all energy ranges is to formulate the theory on a 4-dimensional Euclidean space-time grid, which allows for numerical simulations on state of the art supercomputers. We will review the status of lattice QCD calculations providing examples such as the hadron spectrum and the inner structure of nucleons. We will then point to problems that cannot be solved by conventional Monte Carlo simulation techniques, i.e. chemical potentials and understanding the violation of charge and parity symmetry. It is demonstrated at the example of the Schwinger model that tensor network techniques are able to overcome these problems opening thus a possible path for a solution also in QCD.15.11.17 - Matt Headrick, "Riemannian and Lorentzian flow-cut theorems"
We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut theorem for boundary regions, applied recently to develop a "bit-thread" interpretation of holographic entanglement entropies. We also prove various properties of the max flow and min cut, including respective nesting properties. In the Lorentzian setting, we prove the analogous min flow-max cut theorem, which states that the volume of a maximal slice equals the flux of a minimal flow, where a flow is defined as a divergenceless timelike vector field with norm at least 1. This theorem includes as a special case a continuum version of Dilworth's theorem from the theory of partially ordered sets. We include a brief review of the necessary tools from the theory of convex optimization, in particular Lagrangian duality and convex relaxation.20.10.17 - Sukhi Singh, "Tensor networks, symmetries, and holography"
Ground states of local hamiltonians on a quantum lattice can often be represented efficiently by means of a tensor network. In particular, the ground state of a quantum critical system -- described by a conformal field theory (CFT) in the continuum -- can be efficiently represented by a certain tensor network known as the MERA. According to a recent conjecture, the MERA could provide a discrete realization of at least certain aspects of the AdS/CFT correspondence. Several qualitative similarities between the MERA and the AdS/CFT correspondence have been observed, though how the MERA representation of a critical ground state could possibly encode a dual bulk description in one higher dimension is still an open question. In this talk, I will review the MERA-AdS/CFT conjecture and introduce a candidate bulk description of the MERA. This bulk description has some interesting features, some of which are reminiscent of AdS/CFT. In particular, I will show how this bulk description gauges an onsite global boundary symmetry in the bulk, a general feature of AdS/CFT.19.10.17 - Wilke van der Schee, "Entanglement, MERA and modular hamiltonians"
18.10.17 - Netta Engelhardt, "Decoding the Apparent Horizon: A Coarse-Grained Holographic Entropy"
When a black hole forms from collapse in a holographic theory, the information in the black hole interior remains encoded in the boundary. We prove that the area of the black hole's apparent horizon is precisely the entropy associated to coarse graining over the information in its interior, subject to knowing the exterior geometry. This is the maximum holographic entanglement entropy that is compatible with all classical measurements conducted outside of the apparent horizon. We identify the boundary dual to this entropy and explain why it obeys a Second Law of Thermodynamics.18.10.17 - William Cottrell, "Complexity is Simple"
In this note we investigate the role of Lloyd's computational bound in holographic complexity. Our goal is to translate the assumptions behind Lloyd's proof into the bulk language. In particular, we discuss the distinction between orthogonalizing and `simple' gates and argue that these notions are useful for diagnosing holographic complexity. We show that large black holes constructed from series circuits necessarily employ simple gates, and thus do not satisfy Lloyd's assumptions. We also estimate the degree of parallel processing required in this case for elementary gates to orthogonalize. Finally, we show that for small black holes at fixed chemical potential, the orthogonalization condition is satisfied near the phase transition, supporting a possible argument for the Weak Gravity Conjecture first advocated in Brown et al.16.10.17 - Phuc Nguyen, "Entanglement of purification: from spin chains to holography"
Purification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of many degrees of freedom, one natural purification is the one with minimal entanglement. Here we study the entropy of the minimally entangled purification, called the entanglement of purification, in three model systems: an Ising spin chain, conformal field theories holographically dual to Einstein gravity, and random stabilizer tensor networks. We conjecture values for the entanglement of purification in all these models, and we support our conjectures with a variety of numerical and analytical results. We find that such minimally entangled purifications have a number of applications, from enhancing entanglement-based tensor network methods for describing mixed states to elucidating novel aspects of the emergence of geometry from entanglement in the AdS/CFT correspondence.13.10.17 - Johannes Knaute, "Entanglement Entropy in holographic QCD phase diagrams"
We calculate the holographic entanglement entropy for the holographic QCD phase diagram considered in [arXiv:1702.06731] and explore the resulting qualitative behavior over the temperature-chemical potential plane. In agreement with the thermodynamic result, the phase diagram exhibits the same critical point as the onset of a first-order phase transition curve. We compare the phase diagram of the entanglement entropy to that of the thermodynamic entropy density and find a striking agreement in the vicinity of the critical point. Thus, the holographic entanglement entropy qualifies to characterize different phase structures. The scaling behavior near the critical point is analyzed through the calculation of critical exponents.12.10.17 - Christian Northe, "Topological Complexity in AdS3/CFT2"
We consider subregion complexity within the AdS3/CFT2 correspondence. We rewrite the volume proposal, according to which the complexity of a reduced density matrix is given by the spacetime volume contained inside the associated Ryu-Takayanagi (RT) surface, in terms of an integral over the curvature. Using the Gauss-Bonnet theorem we evaluate this quantity for specific examples. In particular, we find a discontinuity when there is a change in the RT surface, given by a topological contribution. There is no further temperature dependence of the subregion complexity. We propose a CFT expression for this complexity based on kinematic space, and use it to reproduce some of our explicit gravity results obtained at zero temperature.09.10.17 - Lucas Hackl, "Entanglement and Chaos: Linear entropy production at instabilities"
The rate of entropy production in a classical dynamical system is characterized by the Kolmogorov-Sinai entropy rate given by the sum over all positive Lyapunov exponents of the system. I will prove a quantum version of this result valid for bosonic systems with unstable quadratic Hamiltonian: the entanglement entropy of squeezed coherent states grows linearly for large times, with a rate determined by the Lyapunov exponents and the choice of the subsystem. I will discuss its application to quantum field theory and explain our conjecture on what this result implies for quantum chaos and thermalization in systems with periodic orbits.